1. Copyright © 2007 Pearson Education, Inc. Slide 11-1 1, 2, 4, 8, 16 … is an example of a geometric sequence with first term 1 and each subsequent term is 2 times the term preceding it. The multiplier from each term to the next is called the common ratio and is usually denoted by r. 11.3 Geometric Sequences and Series A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant nonzero real number.

2. Copyright © 2007 Pearson Education, Inc. Slide 11-2 11.3 Finding the Common Ratio In a geometric sequence, the common ratio can be found by dividing any term by the term preceding it. The geometric sequence 2, 8, 32, 128, … has common ratio r = 4 since

3. Copyright © 2007 Pearson Education, Inc. Slide 11-3 11.3 Geometric Sequences and Series nth Term of a Geometric Sequence In the geometric sequence with first term a 1 and common ratio r, the nth term a n, is

4. Copyright © 2007 Pearson Education, Inc. Slide 11-4 11.3 Using the Formula for the nth Term Example Find a 5 and a n for the geometric sequence 4, –12, 36, –108, …

5. Copyright © 2007 Pearson Education, Inc. Slide 11-5 11.3 Modeling a Population of Fruit Flies Example A population of fruit flies grows in such a way that each generation is 1.5 times the previous generation. There were 100 insects in the first generation. How many are in the fourth generation.

6. Copyright © 2007 Pearson Education, Inc. Slide 11-6 11.3 Geometric Series A geometric series is the sum of the terms of a geometric sequence. In the fruit fly population model with a 1 = 100 and r = 1.5, the total population after four generations is a geometric series:

7. Copyright © 2007 Pearson Education, Inc. Slide 11-7 11.3 Geometric Sequences and Series Sum of the First n Terms of an Geometric Sequence If a geometric sequence has first term a 1 and common ratio r, then the sum of the first n terms is given by where.

8. Copyright © 2007 Pearson Education, Inc. Slide 11-8 11.3 Finding the Sum of the First n Terms Example Find.